401-4656-21L  Deep Learning in Scientific Computing

SemesterFrühjahrssemester 2021
DozierendeS. Mishra
Periodizitäteinmalige Veranstaltung
LehrspracheEnglisch
KommentarAimed at students in a Master's Programme in Mathematics, Engineering and Physics.



Lehrveranstaltungen

NummerTitelUmfangDozierende
401-4656-21 VDeep Learning in Scientific Computing2 Std.
Mo14-16HG F 5 »
S. Mishra
401-4656-21 UDeep Learning in Scientific Computing1 Std.
Do13-14ML H 44 »
S. Mishra

Katalogdaten

KurzbeschreibungMachine Learning, particularly deep learning is being increasingly applied to perform, enhance and accelerate computer simulations of models in science and engineering. This course aims to present a highly topical selection of themes in the general area of deep learning in scientific computing, with an emphasis on the application of deep learning algorithms for systems, modeled by PDEs.
LernzielThe objective of this course will be to introduce students to advanced applications of deep learning in scientific computing. The focus will be on the design and implementation of algorithms as well as on the underlying theory that guarantees reliability of the algorithms. We will provide several examples of applications in science and engineering where deep learning based algorithms outperform state of the art methods.
InhaltA selection of the following topics will be presented in the lectures.

1. Issues with traditional methods for scientific computing such as Finite Element, Finite Volume etc, particularly for PDE models with high-dimensional state and parameter spaces.

2. Introduction to Deep Learning: Artificial Neural networks, Supervised learning, Stochastic gradient descent algorithms for training, different architectures: Convolutional Neural Networks, Recurrent Neural Networks, ResNets.

3. Theoretical Foundations: Universal approximation properties of the Neural networks, Bias-Variance decomposition, Bounds on approximation and generalization errors.

4. Supervised deep learning for solutions fields and observables of high-dimensional parametric PDEs. Use of low-discrepancy sequences and multi-level training to reduce generalization error.

5. Uncertainty Quantification for PDEs with supervised learning algorithms.

6. Deep Neural Networks as Reduced order models and prediction of solution fields.

7. Active Learning algorithms for PDE constrained optimization.

8. Recurrent Neural Networks and prediction of time series for dynamical systems.

9. Physics Informed Neural networks (PINNs) for the forward problem for PDEs. Applications to high-dimensional PDEs.

10. PINNs for inverse problems for PDEs, parameter identification, optimal control and data assimilation.

All the algorithms will be illustrated on a variety of PDEs: diffusion models, Black-Scholes type PDEs from finance, wave equations, Euler and Navier-Stokes equations, hyperbolic systems of conservation laws, Dispersive PDEs among others.
SkriptLecture notes will be provided at the end of the course.
LiteraturAll the material in the course is based on research articles written in last 1-2 years. The relevant references will be provided.
Voraussetzungen / BesonderesThe students should be familiar with numerical methods for PDEs, for instance in courses such as Numerical Methods for PDEs for CSE, Numerical analysis of Elliptic and Parabolic PDEs, Numerical methods for hyperbolic PDEs, Computational methods for Engineering Applications.

Some familiarity with basic concepts in machine learning will be beneficial. The exercises in the course rely on standard machine learning frameworks such as KERAS, TENSORFLOW or PYTORCH. So, competence in Python is helpful.

Leistungskontrolle

Information zur Leistungskontrolle (gültig bis die Lerneinheit neu gelesen wird)
Leistungskontrolle als Semesterkurs
ECTS Kreditpunkte6 KP
PrüfendeS. Mishra
Formbenotete Semesterleistung
PrüfungsspracheEnglisch
RepetitionRepetition nur nach erneuter Belegung der Lerneinheit möglich.

Lernmaterialien

Keine öffentlichen Lernmaterialien verfügbar.
Es werden nur die öffentlichen Lernmaterialien aufgeführt.

Gruppen

401-4656-21 UDeep Learning in Scientific Computing
GruppeG-01
Do13-14ML H 44 »

Einschränkungen

PlätzeMaximal 200
WartelisteBis 01.03.2021

Angeboten in

StudiengangBereichTyp
Data Science MasterWählbare KernfächerWInformation
Mathematik MasterAuswahl: Numerische MathematikWInformation
Rechnergestützte Wissenschaften MasterWahlfächerWInformation